Numeric.SpecFunctions.Extra:bd0 from math-functions-0.1.5.2

Average Error: 24.05% → 0.31%

Time: 9.6s

Precision: binary64

Cost: 19776

Math

\[x \cdot \log \left(\frac{x}{y}\right) - z \]

↓

\[x \cdot \left(\log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{y}}\right) \cdot 3\right) - z \]

FPCore

(FPCore (x y z) :precision binary64 (- (* x (log (/ x y))) z))

↓

(FPCore (x y z)
 :precision binary64
 (- (* x (* (log (/ (cbrt x) (cbrt y))) 3.0)) z))

C

double code(double x, double y, double z) {
	return (x * log((x / y))) - z;
}

↓

double code(double x, double y, double z) {
	return (x * (log((cbrt(x) / cbrt(y))) * 3.0)) - z;
}

Java

public static double code(double x, double y, double z) {
	return (x * Math.log((x / y))) - z;
}

↓

public static double code(double x, double y, double z) {
	return (x * (Math.log((Math.cbrt(x) / Math.cbrt(y))) * 3.0)) - z;
}

Julia

function code(x, y, z)
	return Float64(Float64(x * log(Float64(x / y))) - z)
end

↓

function code(x, y, z)
	return Float64(Float64(x * Float64(log(Float64(cbrt(x) / cbrt(y))) * 3.0)) - z)
end

Wolfram

code[x_, y_, z_] := N[(N[(x * N[Log[N[(x / y), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]

↓

code[x_, y_, z_] := N[(N[(x * N[(N[Log[N[(N[Power[x, 1/3], $MachinePrecision] / N[Power[y, 1/3], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision] - z), $MachinePrecision]

Target

Original	24.05%
Target	12.45%
Herbie	0.31%

\[\begin{array}{l} \mathbf{if}\;y < 7.595077799083773 \cdot 10^{-308}:\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right) - z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right) - z\\ \end{array} \]

Derivation

1. Initial program 24.05

   \[x \cdot \log \left(\frac{x}{y}\right) - z \]

2. Applied egg-rr 24.12

   \[\leadsto x \cdot \color{blue}{\left(\log \left({\left(\sqrt[3]{\frac{x}{y}}\right)}^{2}\right) + \log \left(\sqrt[3]{\frac{x}{y}}\right)\right)} - z \]

3. Simplified 24.12

   \[\leadsto x \cdot \color{blue}{\left(\log \left(\sqrt[3]{\frac{x}{y}}\right) \cdot 3\right)} - z \]
   Proof

   [Start] 24.12	\[ x \cdot \left(\log \left({\left(\sqrt[3]{\frac{x}{y}}\right)}^{2}\right) + \log \left(\sqrt[3]{\frac{x}{y}}\right)\right) - z \]
   log-pow [=>] 24.12	\[ x \cdot \left(\color{blue}{2 \cdot \log \left(\sqrt[3]{\frac{x}{y}}\right)} + \log \left(\sqrt[3]{\frac{x}{y}}\right)\right) - z \]
   *-lft-identity [<=] 24.12	\[ x \cdot \left(2 \cdot \log \left(\sqrt[3]{\frac{x}{y}}\right) + \color{blue}{1 \cdot \log \left(\sqrt[3]{\frac{x}{y}}\right)}\right) - z \]
   distribute-rgt-out [=>] 24.12	\[ x \cdot \color{blue}{\left(\log \left(\sqrt[3]{\frac{x}{y}}\right) \cdot \left(2 + 1\right)\right)} - z \]
   metadata-eval [=>] 24.12	\[ x \cdot \left(\log \left(\sqrt[3]{\frac{x}{y}}\right) \cdot \color{blue}{3}\right) - z \]

4. Applied egg-rr 0.31

   \[\leadsto x \cdot \left(\log \color{blue}{\left(\sqrt[3]{x} \cdot \frac{1}{\sqrt[3]{y}}\right)} \cdot 3\right) - z \]

5. Simplified 0.31

   \[\leadsto x \cdot \left(\log \color{blue}{\left(\frac{\sqrt[3]{x}}{\sqrt[3]{y}}\right)} \cdot 3\right) - z \]
   Proof

   [Start] 0.31	\[ x \cdot \left(\log \left(\sqrt[3]{x} \cdot \frac{1}{\sqrt[3]{y}}\right) \cdot 3\right) - z \]
   associate-*r/ [=>] 0.31	\[ x \cdot \left(\log \color{blue}{\left(\frac{\sqrt[3]{x} \cdot 1}{\sqrt[3]{y}}\right)} \cdot 3\right) - z \]
   associate-*l/ [<=] 0.31	\[ x \cdot \left(\log \color{blue}{\left(\frac{\sqrt[3]{x}}{\sqrt[3]{y}} \cdot 1\right)} \cdot 3\right) - z \]
   *-rgt-identity [=>] 0.31	\[ x \cdot \left(\log \color{blue}{\left(\frac{\sqrt[3]{x}}{\sqrt[3]{y}}\right)} \cdot 3\right) - z \]

6. Final simplification 0.31

   \[\leadsto x \cdot \left(\log \left(\frac{\sqrt[3]{x}}{\sqrt[3]{y}}\right) \cdot 3\right) - z \]

Alternatives

Alternative 1
Error	12.58%
Cost	26760

\[\begin{array}{l} t_0 := \log \left(\frac{x}{y}\right)\\ t_1 := x \cdot t_0\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;-z\\ \mathbf{elif}\;t_1 \leq 5 \cdot 10^{+301}:\\ \;\;\;\;\mathsf{fma}\left(t_0, x, -z\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{-1}{z}}\\ \end{array} \]

Alternative 2
Error	12.58%
Cost	20424

\[\begin{array}{l} t_0 := x \cdot \log \left(\frac{x}{y}\right)\\ \mathbf{if}\;t_0 \leq -\infty:\\ \;\;\;\;-z\\ \mathbf{elif}\;t_0 \leq 5 \cdot 10^{+301}:\\ \;\;\;\;t_0 - z\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{-1}{z}}\\ \end{array} \]

Alternative 3
Error	9.81%
Cost	13512

\[\begin{array}{l} \mathbf{if}\;x \leq -5.2 \cdot 10^{-195}:\\ \;\;\;\;\mathsf{fma}\left(\log \left(\frac{x}{y}\right), x, -z\right)\\ \mathbf{elif}\;x \leq -5 \cdot 10^{-308}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right) - z\\ \end{array} \]

Alternative 4
Error	0.5%
Cost	13508

\[\begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{-310}:\\ \;\;\;\;x \cdot \left(\log \left(-x\right) - \log \left(-y\right)\right) - z\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\log x - \log y\right) - z\\ \end{array} \]

Alternative 5
Error	33.02%
Cost	6984

\[\begin{array}{l} \mathbf{if}\;z \leq -4.6 \cdot 10^{+21}:\\ \;\;\;\;-z\\ \mathbf{elif}\;z \leq 4.2 \cdot 10^{-68}:\\ \;\;\;\;x \cdot \log \left(\frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \]

Alternative 6
Error	49.51%
Cost	128

\[-z \]

Alternative 7
Error	97.74%
Cost	64

\[z \]

Reproduce

herbie shell --seed 2023088 
(FPCore (x y z)
  :name "Numeric.SpecFunctions.Extra:bd0 from math-functions-0.1.5.2"
  :precision binary64

  :herbie-target
  (if (< y 7.595077799083773e-308) (- (* x (log (/ x y))) z) (- (* x (- (log x) (log y))) z))

  (- (* x (log (/ x y))) z))